*The Department of Mathematics will begin the 2014-2015 Distinguished Lecture Series on November 5-6, 2014 with Professor Emmanuel Candès.*

Professor Emmanuel Candès

Barnum-Simona Chair Professor in Mathematics and Statistics, Stanford University

**Public Lecture**

**Robust Principal Component Analysis**

Wednesday, November 5, 2014

4:00-5:00pm

Natural Sciences I, Room 1114

Reception to Follow

Abstract

This talk is about a curious phenomenon, which concerns the reliable estimation of principal components in the face of severe corruptions. Here, the scientist is given a data matrix which is the sum of an approximately low-rank matrix and a sparse matrix modeling corrupted entries. In addition, many entries may be missing. Hence, we have a blind de-mixing problem in which the goal is to recover the low-rank structure and find out which entries have been corrupted. We present a novel approach to this problem with very surprising performance guarantees as well as a few applications in computer vision and biomedical imaging, where this technique opens new perspectives.

**Mathematics Colloquium****Modern Optimization Meets Physics: Recent Progress on the Phase Retrieval Problem**

Thursday, November 6, 2014

4:00-5:00pm

Natural Sciences II, Room 1201

Reception to Follow

Abstract

In many imaging problems such as X-ray crystallography, detectors can only record the intensity or magnitude of a diffracted wave as opposed to measuring its phase. Phase retrieval concerns the recovery of an image from such phaseless information. Although this problem is in general combinatorially hard, it is of great importance because it arises in many applications ranging from astronomical imaging to speech analysis. This talk discusses novel acquisition strategies and novel convex and non-convex algorithms which are provably exact, thereby allowing perfect phase recovery from a minimal number of noiseless and intensity-only measurements. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. This may be of special contemporary interest because phase retrieval is at the center of spectacular current research efforts collectively known under the name of coherent diffraction imaging aimed, among other things, at determining the 3D structure of large protein complexes.

About Professor Candès

Professor Emmanuel Candès is Barnum-Simons Chair Professor in Mathematics and Statistics at

Stanford University where he received Ph.D. in Statistics in 1998 and began his career as an

assistant professor of Statistics. In 2000, he moved to Caltech, where he was named the Ronald and Maxine Linde Professor of Applied and Computational Mathematics in 2006. He returned to Stanford in 2009. His research includes seminal works and important contributions in signal processing, statistics, scientific computing, optimization and information theory. His early work concerns curvelets and ridgelets to capture geometric structures in signals and images. In 2004, his groundbreaking paper with Terence Tao opened up the field of compressed sensing and its applications. Some of the awards he received include the James H. Wilkinson Prize (2005), the NSF’s Alan.T. Waterman Award (2006), the Information Theory Society Paper Award (2008), the Pólya Prize (2010), the Lagrange Prize (2012), and the D. Heineman Prize of the Academy of Sciences at Göttingen (2013). In 2014 he was elected to the US National Academy of Sciences and American Academy of Arts and Sciences.